The Persistence of Inflation Versus that of Real Marginal Cost in the New Keynesian Model

The Persistence of In?ation Versus that of Real Marginal Cost
in the New Keynesian Model
Julio J. Rotemberg?
Abstract
This note provides an example where the New Keynesian Phillips Curve leads in?ation to
be substantially more persistent than the output gap.
JEL E370
Keywords: In?ation persistence, Phillips curve, New Keynesian Model
As shown by Roberts (1995), several models including Rotemberg (1982) and Calvo
(1983) lead to a ”New Keynesian” Phillips Curve (NKPC) in which current in?ation is related
to the expectation of future in?ation and the current gap between the price charged by the
typical ?rm and the price level that would prevail if they could all charge their preferred
price. It has been suggested (Fuhrer and Moore 1995, Fuhrer 2005) that the NKPC makes
it di?cult for in?ation to be more persistent than the ”output gap” in those cases where
this output gap is perfectly correlated with the gap between the price that ?rms want to
charge and the price that they do charge. This note provides a simple example where the
persistence of in?ation is considerably larger with the hope that this example will help clarify
the qualitative features of this model.
The NKPC is given by
?t = ?mt + ?Et?t+1,
(1)
where Et takes expectations conditional on information available at t, ?t is the in?ation at
t, mt is the gap between the price ?rms would like to charge at t and the price they actually
charge while ? and ? are parameters. Because ? is a discount rate, it is very near though
slightly below one. The variable mt is often referred to as the output gap because output is
higher than ?rms wish it to be when the price they charge is below the one they prefer, i.e.,
?Professor of Business Administration, Harvard Business School, jrotemberg@hbs.edu.
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when mt is positive. In the case where ?rms seek a constant markup over their marginal
cost, it equals (up to constant) the log of their real marginal cost.
Consider ?rst the case where mt follows an AR(1) as in Fuhrer (2005) so that
mt = ?mt?1 + t,
where t is i.i.d. In?ation must then satisfy
?
?t =
m
1 ? ?? t.
(2)
This solution can be veri?ed by noting that (2) implies that
??
Et?t+1 =
m
1 ? ?? t,
so that ?t ? ?Et?t+1 equals ?mt and (1) is satis?ed.
In?ation thus follows the same ?rst order AR process as m and the persistence of the
two variables is the same. However, this result is not robust. Suppose that mt is the sum of
two components
mt = mP + mC,
(3)
t
t
both of which follow AR(1) processes
mi = ?
+ i
i = P, C
t
imit?1
t
where i is i.i.d., i is uncorrelated with j at all leads and lags and the variance of i is ?2.
t
i
The covariance of mt and mt?j is then equal to
?j
Cov
i ?2
i
j(m) =
.
(4)
1 ? ?2
i=P,C
i
The solution for in?ation is now
?
?t =
mi
(5)
1 ? ??
t
i=P,C
i
and this solution can be veri?ed as above by taking the expectation at t of in?ation at t + 1.
The coe?cient on mi in (5) is larger when ?/(1 ? ??
t
i) is larger, i.e., when ?i (which can
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only take values between 0 and 1) is larger. The economic logic for this can be seen in (1).
If in?ation is very persistent (so that Et?t+1 is close to ?t, one will have relatively large
movements in ? for a given movement in m. This means that a given movement in m is
associated with a larger change in in?ation if this movement in m (and thus in ?) is very
persistent.
The autocovariance of in?ation is thus
?j
Cov
i ?2?2
i
j(?) =
.
(1 ? ??
i=P,C
i)2(1 ? ?2
i )
This implies that, relative to its e?ect on the autocovariance of output, the autocovariance
of in?ation is more a?ected by the i whose ?i is larger. This means that the i which has a
more persistent e?ect on output tends to have larger e?ects on the autocovariance of in?ation
so that in?ation tends to become more persistent than output.
Numerically, this e?ect can be quite strong. Suppose that ? = .99, ?C = .7 and ?P = .99
so that this latter component is very persistent. Suppose also that ?2 /?2 = .05(1??2 )/(1?
P
C
P
?2 ) so that the e?ect of the persistent component on the variance of output equals only .05
C
times the e?ect of the “cyclical” component C. Nonetheless these parameters imply that
the e?ect of this persistent component on the variance of in?ation is 39 times larger than
the e?ect of the cyclical component on this variance. The result is a substantial di?erence
between the persistence of in?ation and that of output. The correlation between output at
t and output 8 quarters earlier is only .1 while the correlation of in?ation at t and in?ation
at t ? 8 is .86.
References
Calvo, Guillermo A. (1983) “Staggered Contracts in a Utility Maximizing Framework.” Jour-
nal of Monetary Economics, 12, 383-98.
Fuhrer, Je?rey C. and George R. Moore. (1995) “ In?ation Persistence.” Quarterly Journal
of Economics, 110, 127-59.
Fuhrer, Je?rey C. (2005) “Intrinsic and Inherited In?ation Persistence.” Federal Reserve
Bank of Boston Working Paper 05-8.
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Roberts, John M. (1995) “New Keynesian Economics and the Phillips Curve.” Journal of
Money, Credit and Banking, 27, 975-84.
Rotemberg, Julio J. (1982) “Monopolistic Price Adjustment and Aggregate Output.” Review
of Economic Studies, 49, 517-531.
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